by equating terms with the same order of perturbation from both the sides, where
\(P_0=| \Phi _0 \rangle \langle \Phi _0 |\)  and \(Q_0= 1 - P_0\) . To understand how electron correlation effects propagate from the lower-order level to the higher-order level of perturbation in the determination of polarizabilities, we consider one- and two-orders of \(V_0\)  in the second-order (MBPT(2)) and third-order (MBPT(3)) MBPT method respectively, and estimate the \(\alpha _d\)  and \(\alpha _q\)  values. It is obvious from here that the DF values of the polarizabilities can be obtained by considering zero-order of \(V_0\)  in the calculation. We also intend to verify the results by approximating \(\Omega ^{(0)} \approx \Omega ^{(0,0)} =1\)  and \(\Omega ^{(o,1)} \approx \sum _{n=1}^{\infty } \Omega ^{(n,1)}\)  but accounting only the core-polarization effects in the random-phase approximation (RPA) framework [1]}.
